#9814594

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31)(33,34,39,41)(35,40,44,46)(36,37,42,47)(38,48,45,43);(3,9)(7,13)(10,15)(14,16)(17,19,23,20)(18,24,22,25)(21,28,29,27)(26,32,30,31)(33,35)(34,40)(36,45)(37,43)(38,42)(39,44)(41,46)(47,48);(2,6)(7,13)(8,11)(14,16)(34,41)(35,44)(36,42)(43,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,25,22,24)(21,28,29,27)(26,32,30,31)(33,36,39,42)(34,37,41,47)(35,45,44,38)(40,43,46,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21,23,29)(18,26,22,30)(19,27,20,28)(24,31,25,32)(33,37)(34,42)(35,43)(36,41)(38,40)(39,47)(44,48)(45,46);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)(33,38,39,45)(34,43,41,48)(35,42,44,36)(37,46,47,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9721790);